Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.653351
Title: Structure of scalar-type operators on Lp spaces and well-bounded operators on Hilbert spaces
Author: Khalil, Asma Mohammed
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2002
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Abstract:
It is known that every scalar-type spectral operator on a Hilbert space H is similar to a multiplication operator on some L2 space. The purpose of the main theorem in Chapter 2 of this thesis is to show that every scalar-type spectral operator on an L1 space whose spectral measure has finite multiplicity is similar to a multiplication operator on the same L1 space. Then we prove a similar result for scalar-type spectral operators on Lp (Ω, SΩ, m), p  2, 1 < p < ∞, with spectral measure E(.) of finite uniform multiplicity provided an extra condition is satisfied. Also, we give conditions that make a scalar-type spectral operator on L2(Ω, SΩ, m) similar to a multiplication operator on the same L2(Ω, SΩ, m). In 1954, Dunford proved that a bounded operator T on a Banach space X is spectral if and only if it has the canonical decomposition T = S +Q, where S is a scalar-type operator and Q is a quasinilpotent operator which commutes with S. In Chapter 3, we prove that any well-bounded operator T on a Hilbert space H has the form T = A + Q, where A is a self-adjoint operator and Q is a quasinilpotent operator such that AQ - QA is quasinilpotent. Then we prove that a trigonometrically well-bounded operator T on H can be decomposed as T = U(Q + I) where U is a unitary operator and Q is quasinilpotent such that UQ = QU is also quasinilpotent. In Chapter 4 we prove that an AC-operator with discrete spectrum on H can be decomposed as a sum of a normal operator N and a quasinilpotent Q such that NQ - QN is quasinilpotent. However, the converse of each of the last three theorems is not true in general. In the final chapter we introduce a new class of operators on L2([a,b]) which is larger than the class of well-bounded operators on L2([a,b]) and we call them operators with an AC2-functional calculus. Then we give an example of an operator with an AC2-functional calculus on L2([0,1]) which can be decomposed as a sum of a self-adjoint operator and a quasinilpotent. We also discuss the possibility of decomposing every operator T with an AC2-functional calculus on L2([a,b]) into the sum of a self-adjoint operator A and a quasinilpotent operator Q such that AQ - QA is quasinilpotent.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.653351  DOI: Not available
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